Schur Polynomials and Pl\"ucker degree of Schubert Varieties

Abstract

The polynomial ring $B$ in infinitely many indeterminates $(x_1,x_2,\ldots)$, with rational coefficients, has a vector space basis of Schur polynomials, parametrized by partitions. The goal of this note is to provide an explanation of the following fact. If $\blamb$ is a partition of weight $d$, then the partial derivative of order $d$ with respect to $x_1$ of the Schur polynomial $S_\blamb(\bfx)$ coincides with the Pl\"ucker degree of the Schubert variety of dimension $d$ associated to $\blamb$, equal to the number of standard Young tableaux of shape $\blamb$. The generating function encoding all the degree of Schuberte varieties is determined and some (known) corollaries are also discussed. (The following is an informal report on the contributed talk given by the author during the INPANGA 2020[+1] meeting on Schubert Varieties.)